Saccheri quadrilateral

Builds on Absolute geometry Angle at a point

Definition

Let $A$ , $B$ , $C$ , $D$ be four points of an absolute plane. Assume that $C$ and $D$ lie on the same side of the line $AB$ . The quadrilateral $ABCD$ is a Saccheri quadrilateral with base $AB$ if:

the angles $\angle DAB$ and $\angle ABC$ are right angles;
the legs $AD$ and $BC$ are congruent.

The side $CD$ is the summit, and the angles $\angle BCD$ and $\angle CDA$ are the summit angles.

Remarks

Example: in $\R^2$ , the points $A = (0,0)$ , $B = (2,0)$ , $C = (2,1)$ , $D = (0,1)$ form a Saccheri quadrilateral, since both base angles are right and both legs have length $1$ . Non-example: $A = (0,0)$ , $B = (2,0)$ , $C = (2,1)$ , $D = (0,3)$ has right base angles but legs of lengths $1$ and $3$ , so clause 2 fails. In every absolute plane the summit angles are congruent, and the segment joining the midpoints of base and summit is perpendicular to both. The summit angles are right in the Euclidean plane and acute in the hyperbolic plane.

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Tag	`NRZC1SY4S3`
URI	`m2://NRZC1SY4S3`
Kind	definition
Writing	W0 stub
MSC	51F05, 51M09
Defines	Saccheri quadrilateral
Uses	Absolute geometry, Angle at a point

Saccheri quadrilateral

Definition

Remarks

Reading path

Further reading

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